The analysis of the above reduction combines the arguments from Section REF  with techniques from [1]}. A key difference is that since our test distribution works in the almost uncorrelated regime, we need a more careful completeness analysis i.e,. the reduction from [1]} works in the setting \(\rho \rightarrow 1\)  whereas our tests are based in the setting \(\rho \rightarrow 0\) . Another additional component in our proof is that is that we strengthen \(k\) -ary noise stability estimate for functions over the \([R]\) -ary hypercube (i.e, (REF )) with small low degree influences to the setting of arbitrary probability spaces.
